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Tensor computations are increasingly prevalent numerical techniques in data science, but pose unique challenges for high-performance implementation. We provide novel algorithms and systems infrastructure which enable efficient parallel implementation of algorithms for tensor completion with generalized loss functions. Specifically, we consider alternating minimization, coordinate minimization, and a quasi-Newton (generalized Gauss-Newton) method. By extending the Cyclops library, we implement all of these methods in high-level Python syntax. To make possible tensor completion for very sparse tensors, we introduce new multi-tensor primitives, for which we provide specialized parallel implementations. We compare these routines to pairwise contraction of sparse tensors by reduction to hypersparse matrix formats, and find that the multi-tensor routines are more efficient in theoretical cost and execution time in experiments. We provide microbenchmarking results on the Stampede2 supercomputer to demonstrate the efficiency of the new primitives and Cyclops functionality. We then study the performance of the tensor completion methods for a synthetic tensor with 10 billion nonzeros and the Netflix dataset, considering both least squares and Poisson loss functions. • Novel algorithms that use second information for generalized tensor completion. • Novel formulation of quasi-Newton algorithm for generalized tensor completion. • New programming abstractions in Python for sparse tensor kernels. • First parallel implementation of generalized tensor completion algorithms. [ABSTRACT FROM AUTHOR]